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Register a new userA translation of L. Eulers On the motion of comets in parabolic orbits, having the Sun in the focus
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Sylvio R Bistafa
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This is a translation from Latin of E840 De motu cometarum in orbitis parabolicis, solem in foco habentibus, in which Euler addresses six problems related to comets in heliocentric parabolic orbits. Problem 1: Find the true anomaly of a heliocentric comet from the latus rectum of the orbit and the medium Earth to Sun distance. Problem 2: Find the orbit of a heliocentric comet from three given positions. Problem 3: Knowing the orbit of a comet, and the instant in time in which it dwells in the perihelion, define its longitude and latitude at any time. Problem 4: From two locations of a heliocentric comet, find the inclination of the comets orbit in relation to the ecliptic, and the positions of the nodes. Problem 5: From the time before or after the comet had reached the perihelion, and from the comets distance to the perihelion as seen from the Sun, find the same distance in another time before or after it had appeared in the perihelion. Problem 6: Find the orbit of a comet from three given heliocentric longitudes and latitudes. From these problems, several corollaries and scholia are derived.
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This is a translation from Latin of E348 Methodus facilis motus corporum coelestium utcunque perturbatos ad rationem calculi astronomici revocandi, in which Euler develops a method to alleviate the astronomical computations in a typical celestial three-body problem represented by Sun, Earth and Moon. In this work, Eulers approach consists of two parts: geometrical and mechanical. The geometrical part contains most of the analytical developments, in which Euler makes use of Cartesian and spherical trigonometry as well - the latter not always in a clear enough way. With few sketches to show the geometrical constructions envisaged by Euler - represented by several geometrical variables -, it is a hard to follow publication. The Translator, on trying to clear the way to the non-specialized reader, used the best of his abilities to add his own figures to the translation. In the latter part of the work, Euler particularizes his developments to the Moon, ending up with eight coupled differential equations for resolving the perturbed motion of this celestial body, which makes his claim of an easy method as being rather fallacious. Despite showing great analytical skills, Euler did not give indications on how this system of equations could be solved, which renders his efforts practically useless in the determination of the variations of the nodal line and inclination of the Moons orbit.
In this paper, we discuss the work on comets of Caroline Herschel, the first female comet-hunter. After leaving Bath for the environs of Windsor Castle and eventually Slough, she discovered at least eight comets, five of which were reported in the Philosophical Transactions of the Royal Society. We consider her public image, astronomers perceptions of her contributions, and the style of her astronomical drawings that changed with the technological developments in astronomical illustration.
This is an English translation of E579 in which the introductory remarks are in French, while Eulers original text is in Latin. By considering the balance of forces acting on a raising balloon on an isothermal atmosphere, namely the weight of the balloon, the buoyant force, and the aerodynamic drag force, Euler provides closed formulas for the calculation of the maximum altitude reached by the balloon, the altitude for which the velocity is maximum, the maximum velocity attained by the balloon, and the total ascending time.
The gravitational field of the sun will focus light from a distant source to a focal point at a minimal distance of 550 Astronomical Units from the sun. A proposed mission to this gravitational focus could use the sun as a very large lens, allowing (in principle) a large amplification of signal from the target, and a very high magnification. This article discusses some of the difficulties involved in using the sun as such a gravitational telescope for a candidate mission, that of imaging the surface of a previously-detected exoplanet. These difficulties include the pointing and focal length, and associated high magnification; the signal to noise ratio associated with the solar corona, and the focal blur. In addition, a method to calculate the signal gain and magnification is derived using the first-order deflection calculation and classical optics, showing that the gain is finite for an on-axis source of non-zero area.
The long-term dynamics of Oort cloud comets are studied under the influence of both the radial and the vertical components of the Galactic tidal field. Sporadic dynamical perturbation processes are ignored, such as passing stars, since we aim to study the influence of just the axisymmetric Galactic tidal field on the cometary motion and how it changes in time. We use a model of the Galaxy with a disc, bulge and dark halo, and a local disc density, and disc scale length constrained to fit the best available observational constraints. By integrating a few million of cometary orbits over 1 Gyr, we calculate the time variable flux of Oort cloud comets that enter the inner Solar System, for the cases of a constant Galactic tidal field, and a realistically varying tidal field which is a function of the Suns orbit. The applied method calculates the evolution of the comets by using first-order averaged mean elements. We find that the periodicity in the cometary flux is complicated and quasi-periodic. The amplitude of the variations in the flux are of order 30%. The radial motion of the Sun is the chief cause of this behaviour, and should be taken into account when the Galactic influence on the Oort cloud comets is studied.
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